Many systems have circuit implementations of a linear transformation, such as discrete Fourier transform (DFT) and/or an inverse discrete Fourier transform (IDFT). For example, communications systems that utilize multi-tone links often implement the IDFT during transmission of data and the DFT during receiving of the data. These transformations are useful in getting close to capacity from the communication channel.
The DFT and/or the IDFT are often implemented using digital circuits. This is illustrated by circuits 100 and 150 shown in FIGS. 1A and 1B, respectively. The circuits 100 and 150 may be included in transmitters and receivers in communication systems. In FIG. 1A, the circuit 100 may include an IDFT, which transforms an input vector X 110 into intermediate output Y=FX 114, and parallel-to-serial (P/S) converter 116, which converts the intermediate output Y=FX 114 into a serial data stream. Typically, these operations are implemented in a digital domain 122. A digital-to-analog converter (DAC) 118 converts digital signals to an analog domain 124 yielding an output V 120. The IDFT 112 and the DAC 118 each may be clocked at a rate that is at least at the Nyquist rate (two times the symbol rate) using a clock 126.
In FIG. 1B, the circuit 150 may receive input signal 152. The input signal 152 may be the output V 120. The input signal 152 is converted from the analog domain 124 to the digital domain 122 by analog-to-digital converter (ADC) 154. The circuit 150 may include serial-to-parallel (S/P) converter 156 and a DFT 158 to convert the digital signals to a vector V 162. The ADC 154 and the DFT 158 each may be clocked at least at the Nyquist rate using a clock 160. At high data rates, however, circuits, such as the circuit 100 (FIG. 1A) and the circuit 150, may have excessive sampling rates, i.e., high frequencies for the clocks 126 (FIG. 1A) and 160, and resolution or quantization requirements. As a consequence, digital implementations of transformations such as the IDFT 112 and the DFT 158, may be complex, costly and may consume significant amounts of power. There is a need, therefore, for improved linear transformation circuits.
Like reference numerals refer to corresponding parts throughout the drawings.